cvpr cvpr2013 cvpr2013-421 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Peng Wang, Jingdong Wang, Gang Zeng, Weiwei Xu, Hongbin Zha, Shipeng Li
Abstract: In visual recognition tasks, the design of low level image feature representation is fundamental. The advent of local patch features from pixel attributes such as SIFT and LBP, has precipitated dramatic progresses. Recently, a kernel view of these features, called kernel descriptors (KDES) [1], generalizes the feature design in an unsupervised fashion and yields impressive results. In this paper, we present a supervised framework to embed the image level label information into the design of patch level kernel descriptors, which we call supervised kernel descriptors (SKDES). Specifically, we adopt the broadly applied bag-of-words (BOW) image classification pipeline and a large margin criterion to learn the lowlevel patch representation, which makes the patch features much more compact and achieve better discriminative ability than KDES. With this method, we achieve competitive results over several public datasets comparing with stateof-the-art methods.
Reference: text
sentIndex sentText sentNum sentScore
1 The advent of local patch features from pixel attributes such as SIFT and LBP, has precipitated dramatic progresses. [sent-2, score-0.124]
2 Recently, a kernel view of these features, called kernel descriptors (KDES) [1], generalizes the feature design in an unsupervised fashion and yields impressive results. [sent-3, score-0.6]
3 In this paper, we present a supervised framework to embed the image level label information into the design of patch level kernel descriptors, which we call supervised kernel descriptors (SKDES). [sent-4, score-1.046]
4 Specifically, we adopt the broadly applied bag-of-words (BOW) image classification pipeline and a large margin criterion to learn the lowlevel patch representation, which makes the patch features much more compact and achieve better discriminative ability than KDES. [sent-5, score-0.389]
5 1), much emphasis has been directed at coding the patch descriptors such as soft coding [24], sparse coding [5, 10, 37], building a discriminative dictionary [13, 26, 43] and discovering good pooling strategies [8] and receptive fields [3, 12]. [sent-12, score-0.747]
6 Nevertheless, most work keeps the low level descriptors with our supervised kernel descriptors (SKDES). [sent-13, score-0.704]
7 As elaborated by [27, 36], the selection of raw descriptors is also an essential factor for achieving good performance in recognition tasks as the error at the beginning may propagate to latter stages. [sent-17, score-0.162]
8 In this work, we focus on learning discriminative patch descriptors by exploiting image label information for improving racognition accuracy. [sent-18, score-0.32]
9 Related work In recent years, the research over designing local descriptors has attracted much attention since the success of SIFT and HOG in retrieval, detection and recognition. [sent-21, score-0.162]
10 In terms of the orientation histogram, a bunch of data-driven descriptors are proposed. [sent-22, score-0.203]
11 [7] used spherical k-means to get dense data-driven filters for local descriptors and showed better recognition accuracy than HOG on the PASCAL challenge. [sent-25, score-0.196]
12 Additionally, some works tried to adopt supervised techniques such as the linear discriminant analysis (LDA), the local discriminant embedding (LDE) to pursue a set of projections that best separate descriptors of different classes. [sent-27, score-0.288]
13 [3 1] tried to learn a non-linear transformation with deep networks by minimizing a margin-based cost function and presented impressive results on object retrieval tasks. [sent-32, score-0.123]
14 [35] provided a supervised compact patch descriptor by solving convex max-margin problems and utilized a sparse and low-rank regularization to conduct pooling region selection and supervised dimension reduction. [sent-34, score-0.725]
15 However, these supervised approaches must have the label information associated with each pair of image patches, i. [sent-35, score-0.126]
16 How to learn discriminative low level descriptors for image recognition is rarely addressed. [sent-39, score-0.231]
17 [16] applied a large, deep convolutional neural network with the hidden unit dropout regularization over the ImageNet recognition challenges1 and obtained the best performance. [sent-44, score-0.196]
18 Recently, to avoid labeling training data, deep learning methods utilized unsupervised fashion such as convolutional deep belief networks [20], convolutional sparse coding [14] and deconvolutional networks [44]. [sent-46, score-0.498]
19 While in most cases like deep belief networks, the results can be further improved by exploiting supervised label information. [sent-47, score-0.195]
20 In this work, we focus on using image level label information to guide the design of low level features by taking advantage of kernel descriptors (KDES) [1]. [sent-48, score-0.451]
21 Based on such a view, it provides a unified way to generate low-level patch features from multiple pixel attributes (gradient, color etc. [sent-51, score-0.124]
22 While kernel descriptors are good at representing raw pixel features for recognition, it generates very high dimen1http://www. [sent-54, score-0.381]
23 org/challenges/LSVRC/2012 sional features because of the kernel trick for explicit representation and Kronecker production for combining multiple features. [sent-56, score-0.219]
24 Though the original work applied the kernel principal component analysis (KPCA) to learn a compact representation, it loses the discriminative ability when dropping to a relatively low dimensional space. [sent-57, score-0.319]
25 In our work, a supervised model, the large margin nearest neighbor (LMNN) criterion [38], is investigated to learn low dimensional patch-level kernel descriptors which we call supervised kernel descriptors (SKDES). [sent-58, score-1.089]
26 Preliminaries Kernel descriptors [1] highlight the kernel view of orientation histograms and show that descriptors like SIFT and HOG are a particular type of match kernels over patches. [sent-61, score-0.584]
27 A patch is then represented by a histogram zo)f· o·r·iehnted gradients, aggregated over the pixels, where m˜ (z) Fh(P) =? [sent-64, score-0.124]
28 Kernel descriptors In KDES, the linear kernel of orientation histogram in Eqn. [sent-80, score-0.422]
29 ), then the similarity between ori- ented gradients can be viewed in a kernel form, Kh(P,Q) = ? [sent-86, score-0.219]
30 ∈Q Furthermore, the kernel can be generalized to capture the positions of pixels, Kgrad(P,Q) = ? [sent-95, score-0.219]
31 Then, 222888555977 the patch feature can be viewed as a kernel descriptor, Fgrad(P) = ? [sent-110, score-0.343]
32 Compact kernel descriptors To make the computation efficient and the representation compact, a kernel dimension reduction approach is presented in [1]. [sent-115, score-0.6]
33 Given a set of densely sampled basis vectors, such as the sampled orientations {φp(yj)}jd=p1, {φo(xi)}id=o1 × and the sampled position vectors the gradient kernel descriptors are approximated in t)h}e form where the t-th component is written as follows, F¯gtrad(P) =? [sent-118, score-0.423]
34 P Here {αtij } forms the tth kernel principal componentH tehraet {isα le}arn feodrm over ethe t joint basis vectors, {φo(x1) ⊗ φp (hay1t t) i , s· ·l ·e , nφoe (dx odov )e ⊗φp(ydp )n }t. [sent-129, score-0.219]
35 =re cλistαelyt,, nw KheDreE SK{c αis a centralized kernel matrix which is computed from the joint basis vectors. [sent-131, score-0.256]
36 In this process, the kernel descriptors are explicitly represented through the kernel trick and Kronecker product, and finally compressed by using KPCA. [sent-132, score-0.6]
37 We aim to learn compact kernel descriptors by exploiting the image labels, i. [sent-136, score-0.447]
38 Then generally, the patch feature can be writt,en·· as Fα(P) = ATk, where A is called a kernel transform which is a D D matrix and D is the dimensionality orfm a wkehirnchel descriptor Dk . [sent-143, score-0.476]
39 , Pooling the patch features in region Rs together yields the feature over a region, fRs = op|hR=s1| g(H, ATkh), (4) where |Rs | is the patch feature number inside the region Rwhs,e op iRs a pooling operator, e. [sent-145, score-0.422]
40 ϕ(H, A; I), where RN is the region number and I an is image that can be represented from the kernel features k. [sent-150, score-0.219]
41 The goal we are interested in is to find the compact kernel transform A through a supervised technique, the large margin nearest neighbor criterion [38] specifically. [sent-151, score-0.486]
42 A sample and its target neighbors (one of its k-nearest neighbors that share the same label) are set to be close while other samples with different labels are pushed farther than the kth target neighbor by a large margin. [sent-152, score-0.168]
43 Two constraints are introduced, the first term is a large margin penalty that penalizes small distances between each input and all other inputs that do not share the same label, while the second term penalizes large distances between each input and its target neighbors. [sent-153, score-0.16]
44 In addition, to avoid over-fitting issues and make the descriptor compact, we induce a rank regularization term to control the complexity of the model. [sent-154, score-0.21]
45 rforms a convex surrogate of rank(A) which encourages low rank of the transformation. [sent-174, score-0.166]
46 (4) encodes a patch feature ATkh with a pre-computed dictionary through ridge regression which is defined as follows, ch∗ = argminch ? [sent-178, score-0.227]
47 (6) Then the code ch has a closed-form solution respecting the patch feature and the dictionary as follows, ch∗ = (HTH + μI)−1HT(ATkh), (7) where I the identity matrix. [sent-183, score-0.263]
48 The dictionary H is defined in the kernel space, i. [sent-185, score-0.322]
49 Specifically, given the set of ˜ p patch level kernel features F (one column is a kernel descriptor k), the dictionary of size n can be represented as H = (ATF)ZT, where Z is a combination matrix of size n ˜ p which clustwerhse n w Zo risds a o coutm obfi ˜n p patches. [sent-189, score-0.784]
50 222888556088 With the encoded patch features, supposing the spatial pooling operation op in Eqn. [sent-190, score-0.298]
51 (4) is an average pooling operation, and the image feature is concatenated from different regions, an image feature can be represented as: = ? [sent-191, score-0.135]
52 h|R=s1|khi, + μI)−1, (8) where khi is the h-th patch inside region Rs of the i-th image. [sent-203, score-0.172]
53 (5) regarding A and Z, which can be reduced to computing the gradient of dij w. [sent-205, score-0.187]
54 Optimization × Directly applying the gradient decent algorithm regarding the optimizing parameters is difficult, because of the complexity from computing the gradient of high order matrix and the inverse in Eqn. [sent-210, score-0.193]
55 (8), the image features are computed through a linear mapping L from the pooled kernel descriptors, and the distance can be written in the following form, dij = (ksi − ksj)TM(ksi − ksj) by setting M = LLT w? [sent-217, score-0.326]
56 3 that our descriptors can be represented using the learned matrix M. [sent-222, score-0.199]
57 As B is a D n matrix, we further know rank(B) =A D B by assuming tmhaatthe dictionary H for coding is over-complete, which is always the case for the BOW pipeline. [sent-227, score-0.166]
58 sR=N1 can be equivalently transformed into a rank constraint on M. [sent-230, score-0.114]
59 tT=1f(w,zt) + R(w), (10) where w is the weight vector to be learned, zt is the tth available training sample (pair), and f(w, zt) is a convex loss, and R(w) is a convex regularization term. [sent-258, score-0.223]
60 At iteration t, RDA uses the loss subgradient gt ∈ ∂f(wt, zt) to perform the update, wt+1= argmwin? [sent-261, score-0.115]
61 (1 + dij − dil)), where we use the smooth hinge loss, h(x) = b1 log(1 +ebx), for computing the subgradient of hinge loss at 0, and we set b = 10. [sent-288, score-0.27]
62 Kernel of kernel Having M at hand, we avoid the difficulty to recover A and Z in Eqn. [sent-292, score-0.219]
63 (8) by utilizing the learned distances to represent kernel similarity of encoded patches descriptors through the efficient match kernel method (EMK) [2]. [sent-293, score-0.679]
64 In practice, we construct the RBF kernel between a pair of encoded patch features in Eqn. [sent-294, score-0.343]
65 Through this kernel view, the final supervised kernel descriptors (SKDES) f can be represented efficiently by decomposing the M into i. [sent-310, score-0.726]
66 , f = where is injective (a column full rank matrix) with rank of d, d ? [sent-312, score-0.228]
67 Parameter setting In our experiments, for constructing raw kernel descriptors, we adopt the code provided by Bo et al. [sent-331, score-0.219]
68 We set the patch size to 16 16, the sampling step length to 8 pixels and rescale the 1m6a×xi1m6u, mth length oinfg images ntog t3h0 0to pixels. [sent-335, score-0.124]
69 We set the default codebook size (number of words in the dictionary C of Sec. [sent-337, score-0.165]
70 We use the spatial pyramid [18] for pooling, set the pyramid level to 1 1, 2 2, 3 3 and 4p ×o i4n ga,n dse weight trhaem icdel ll s alt layer ×l t,o 2 2l1 × fo 2r, measuring t4he × importance hoft tthhee cfeealltu sre ast ilany ceerl ll s. [sent-340, score-0.137]
71 We evaluate the performances of each descriptor and the combination of the three kernel descriptors by simply concatenating the image-level feature vectors. [sent-344, score-0.428]
72 5 and we investigate two influential parameters which are λ∗ for the rank regularization and γm for efficient match kernel over a validation image set. [sent-348, score-0.382]
73 We enumerate the parameters over the range from 10−3 to 102, and the results from the UIUC sport dataset are showed in Fig. [sent-349, score-0.118]
74 dimensionality To demonstrate that our supervised method can learn more compact low level descriptors, we tested the performance of our supervised descriptor under different dimensionality over UIUC sport. [sent-358, score-0.498]
75 3 shows the evaluation results of our supervised dimensionality reduction using the compact mapping Lˆ from Sec. [sent-360, score-0.241]
76 Classification accuracy comparison between the baseline KDES [1] and our approach in dimension reduction of patch descriptors. [sent-372, score-0.124]
77 ments show the gain from supervised information is consistent between different codebooks and grid sizes in Sec. [sent-382, score-0.126]
78 The supervised scheme performs very well, as there is less drop in accuracy going from 300 dimensions to 10. [sent-385, score-0.126]
79 This shows the supervised strategy greatly improves storage and computational efficiency of the kernel descriptors. [sent-388, score-0.345]
80 codebook and pyramid size To empirically justify the performance gain from the supervised information, we trained dictionaries of multiple codebook sizes, constructed spatial pooling pyramids of different levels and compared the accuracy produced from KDES and SKDES in Fig. [sent-394, score-0.436]
81 As can be observed, the gain from SKDES is consistently obtained over different codebook sizes and different spatial pyramid sizes, which means that the supervised information is complementary to other stages of the classification pipeline. [sent-396, score-0.239]
82 Another observation is that the performance is not saturated at the largest dictionary we had tested, and by combining with word selection strategies, such as the one in [12], a compact dictionary can be selected from orginal large one, which yields further improvements with fewer words. [sent-397, score-0.272]
83 UIUC sport [22] contains 1579 images with 8 sport event categories. [sent-401, score-0.168]
84 021 (a) Images belonging to boc e misclas if ed as croquet (b) Images belonging to croquet misclassified as bocce Figure 5. [sent-411, score-0.266]
85 Up: The confusion matrix over UIUC sport dataset (%). [sent-412, score-0.167]
86 09741265 (a) Images belonging to MITcoast misclas if ed as MITopencountry (b) Images belonging to MITopencountry misclassified as MITcoast Figure 6. [sent-451, score-0.115]
87 Moreover, the confusion matrix and misclassified images are showed in Fig. [sent-458, score-0.184]
88 The winner [8] proposed a geometrically discriminative pooling strategy that can also be regarded as a complementary strategy for our work. [sent-504, score-0.201]
89 Effectiveness of learned feature distances We finally empirically show that our learned distances generate more discriminative features. [sent-507, score-0.12]
90 7, we compared the RBF kernel matrix, between testing and training data of UIUC sport, generated from the image features in Sec. [sent-509, score-0.219]
91 As can be seen, SKDES provides much more distinctive kernel similarity (much clearer diagonal structure of the matrix), which effectively pushes the images with different labels farther and pulls the images with the same label closer. [sent-513, score-0.251]
92 Conclusion In this paper, we proposed supervised kernel descriptors (SKDES) for local image patches, which is learned from image labels based on the large margin criterion with low-rank regularization and the widely-applied BOW image classification pipeline. [sent-517, score-0.597]
93 (a) The kernel matrix between training and testing image features from KDES. [sent-554, score-0.256]
94 (b) The kernel matrix from our SKDES, which yields a much clearer diagonal structure indicating the kernel similarity from SKDES is more discriminative. [sent-556, score-0.507]
95 Efficient match kernel between sets of features for visual recognition. [sent-572, score-0.219]
96 Ask the locals: Multi-way local pooling for image recognition. [sent-582, score-0.135]
97 Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations. [sent-706, score-0.123]
98 Multi-channel shape-flow kernel descriptors for robust video event detection and retrieval. [sent-777, score-0.381]
99 Linear spatial pyramid matching using sparse coding for image classification. [sent-879, score-0.114]
100 Adaptive deconvolutional networks for mid and high level feature learning. [sent-894, score-0.122]
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simIndex simValue paperId paperTitle
1 0.92018414 141 cvpr-2013-Efficient Computation of Shortest Path-Concavity for 3D Meshes
Author: Henrik Zimmer, Marcel Campen, Leif Kobbelt
Abstract: In the context of shape segmentation and retrieval object-wide distributions of measures are needed to accurately evaluate and compare local regions ofshapes. Lien et al. [16] proposed two point-wise concavity measures in the context of Approximate Convex Decompositions of polygons measuring the distance from a point to the polygon ’s convex hull: an accurate Shortest Path-Concavity (SPC) measure and a Straight Line-Concavity (SLC) approximation of the same. While both are practicable on 2D shapes, the exponential costs of SPC in 3D makes it inhibitively expensive for a generalization to meshes [14]. In this paper we propose an efficient and straight forward approximation of the Shortest Path-Concavity measure to 3D meshes. Our approximation is based on discretizing the space between mesh and convex hull, thereby reducing the continuous Shortest Path search to an efficiently solvable graph problem. Our approach works outof-the-box on complex mesh topologies and requires no complicated handling of genus. Besides presenting a rigorous evaluation of our method on a variety of input meshes, we also define an SPC-based Shape Descriptor and show its superior retrieval and runtime performance compared with the recently presented results on the Convexity Distribution by Lian et al. [12].
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3 0.87191135 79 cvpr-2013-Cartesian K-Means
Author: Mohammad Norouzi, David J. Fleet
Abstract: A fundamental limitation of quantization techniques like the k-means clustering algorithm is the storage and runtime cost associated with the large numbers of clusters required to keep quantization errors small and model fidelity high. We develop new models with a compositional parameterization of cluster centers, so representational capacity increases super-linearly in the number of parameters. This allows one to effectively quantize data using billions or trillions of centers. We formulate two such models, Orthogonal k-means and Cartesian k-means. They are closely related to one another, to k-means, to methods for binary hash function optimization like ITQ [5], and to Product Quantization for vector quantization [7]. The models are tested on largescale ANN retrieval tasks (1M GIST, 1B SIFT features), and on codebook learning for object recognition (CIFAR-10). 1. Introduction and background Techniques for vector quantization, like the well-known k-means algorithm, are used widely in vision and learning. Common applications include codebook learning for object recognition [16], and approximate nearest neighbor (ANN) search for information retrieval [12, 14]. In general terms, such techniques involve partitioning an input vector space into multiple regions {Si}ik=1, mapping points x in each region uinlttiop region-specific representatives {ci}ik=1, known as cioennte irnst.o A resg siuonch-,s a quantizer, qse(nxt)a,t can b {ec expressed as k q(x) = ?1(x∈Si)ci, (1) i?= ?1 where 1(·) is the usual indicator function. eTrhee 1 quality oef u a quantizer oisr expressed in terms of expected distortion, a common measure of which is squared error ?x − q(x) ?22. In this case, given centers {ci}, the region t ?ox xw −hic qh(x a point nis t assigned gwivitehn m ceinnitemrasl {dcis}to,r tthioen r eisobtained by Euclidean nearest neighbor (NN) search. The k-means algorithm can be used to learn centers from data. To reduce expected distortion, crucial for many applications, one can shrink region volumes by increasing k, the number of regions. In practice, however, increasing k results in prohibitive storage and run-time costs. Even if one resorts to ANN search with approximate k-means [14] or hierarchical k-means [12], it is hard to scale to large k (e.g., k = 264), as storing the centers is untenable. This paper concerns methods for scalable quantization with tractable storage and run-time costs. Inspired by Product Quantization (PQ), a state-of-the-art algorithm for ANN search with high-dimensional data (e.g., [7]), compositionality is one key. By expressing data in terms of recurring, reusable parts, the representational capacity of compositional models grows exponentially in the number ofparameters. Compression techniques like JPEG accomplish this by encoding images as disjoint rectangular patches. PQ divides the feature space into disjoint subspaces that are quantized independently. Other examples include part-based recognition models (e.g., [18]), and tensor-based models for stylecontent separation (e.g., [17]). Here, with a compositional parameterization of region centers, we find a family of models that reduce the enc√oding cost of k centers down from k to between log2 k and √k. A model parameter controls the trade-off between model fidelity and compactness. We formulate two related algorithms, Orthogonal kmeans (ok-means) and Cartesian k-means (ck-means). They are natural extensions of k-means, and are closely related to other hashing and quantization methods. The okmeans algorithm is a generalization of the Iterative Quantization (ITQ) algorithm for finding locality-sensitive binary codes [5]. The ck-means model is an extension of okmeans, and can be viewed as a generalization of PQ.1 We then report empirical results on large-scale ANN search tasks, and on codebook learning for recognition. For ANN search we use datasets of 1M GIST and 1B SIFT features, with both symmetric and asymmetric distance measures on the latent codes. We consider codebook learning for a generic form of recognition on CIFAR-10. 2. k-means Given a dataset of n p-dim points, D ≡ {xj }jn=1, the kmeans algorithm partitions ithme n points in ≡to { kx c}lusters, and 1A very similar generalization of PQ, was developed concurrently by Ge, He, Ke and Sun, and also appears in CVPR 2013 [4]. 333000111755 represents each cluster by a center point. Let C ∈ Rp×k breep a mseantrtsix e wachhos celu sctoelurm byns a comprise tihnet .k Lceltus Cter ∈ centers, i.e., C = [c1, c2 , · · · , ck] . The k-means objective is to minimize the within,-c··lu·s ,tecr squared distances: ?k-means(C) = = x?∈Dmiin?x − ci?22 x?∈Db∈mHin1/k?x − Cb?22 (2) where H1/k ≡ {b | b ∈ {0, 1}k and ?b? = 1}, i.e., b is a binary vee Hctor comprising a ,1-1o}f-ka encoding. Lloyd’s b k- imse aa bni-s algorithm [11] finds a local minimum of (2) by iterative, alternating optimization with respect to C and the b’s. The k-means model is simple and intuitive, using NN search to assign points to centers. The assignment of points to centers can be represented with a log k-bit index per data point. The cost of storing the centers grows linearly with k. 3. Orthogonal k-means with 2m centers With a compositional model one can represent cluster centers more efficiently. One such approach is to re- construct each input with an additive combination of the columns of C. To this end, instead of the 1-of-k encoding in (2), we let b be a general m-bit vector, b ∈ Hm ≡ {0, 1}m, (a2n)d, wCe e∈ l Rt bp× bme. aA gse nsuerchal, meac-bhi tc vluesctteorr c, ebnt ∈er H His th≡e sum o}f a asundbs Cet o∈f Rthe columns of C. There are 2m possible subsets, and therefore k = 2m centers. While the number of parameters in the quantizer is linear in m, the number of centers increases exponentially. While efficient in representing cluster centers, the approach is problematic, because solving bˆ = abrg∈Hmmin?x − Cb?22,(3) is intractable; i.e., the discrete optimization is not submodular. Obviously, for small 2m one could generate all possible centers and then perform NN search to find the optimal solution, but this would not scale well to large values of m. One key observation is that if the columns of C are orthogonal, then optimization (3) becomes tractable. To explain this, without loss of generality, assume the bits belong to {−1, 1} instead of {0, 1}, i.e., b? ∈ Hm? ≡ {−1, 1}m. tToh e{−n, ?x − Cb??22 = xTx + b?TCTCb? − 2xTCb? .(4) For diagonal CTC, the middle quadratic term on the RHS becomes trace(CTC), independent of b?. As a consequence, when C has orthogonal columns, abr?g∈mH?min?x − Cb??22 = sgn(CTx) ,(5) where sgn(·) is the element-wise sign function. Cluster centers are depicted by dots, and cluster boundaries by dashed lines. (Left) Clusters formed by a 2-cube with no rotation, scaling or translation; centers = {b? |b? ∈ H2? }. (Right) Rotation, scaling and translation are used to reduce distances between data points and cluster centers; centers = {μ + RDb? | b? ∈ H2? }. To reduce quantization error further we can also introduce an offset, denoted μ, to translate x. Taken together with (5), this leads to the loss function for the model we call orthogonal k-means (ok-means):2 ?ok-means(C,μ) = x?∈Db?m∈iHn?m?x − μ − Cb??22. (6) Clearly, with a change of variables, b? = 2b −1, we can define new versions of μ and C, with ide=ntic 2abl −lo1s,s, w wfoer c wanh dicehthe unknowns are binary, but in {0, 1}m. T uhnek nook-wmnesa anrse quantizer uetn icnod {e0,s 1ea}ch data point as a vertex of a transformed m-dimensional unit hypercube. The transform, via C and μ, maps the hypercube vertices onto the input feature space, ideally as close as possible to the data points. The matrix C has orthogonal columns and can therefore be expressed in terms of rotation and scaling; i.e., C ≡ RD, where R ∈ Rp×m has orthonormal cinoglu;m i.ne.s, ( CRT ≡R = R DIm,) w, ahnerde eD R Ris ∈ diagonal and positive definite. The goal of learning is to find the parameters, R, D, and μ, which minimize quantization error. Fig. 1 depicts a small set of 2D data points (red x’s) and two possible quantizations. Fig. 1 (left) depicts the vertices of a 2-cube with C = I2 and zero translation. The cluster regions are simply the four quadrants of the 2D space. The distances between data points and cluster centers, i.e., the quantization errors, are relatively large. By comparison, Fig. 1 (right) shows how a transformed 2-cube, the full model, can greatly reduce quantization errors. 3.1. Learning ok-means To derive the learning algorithm for ok-means we first rewrite the objective in matrix terms. Given n data points, let X = [x1, x2, · · · , xn] ∈ Rp×n. Let B? ∈ {−1, 1}m×n denote the corresponding ∈clu Rster assignment {co−ef1f,ic1i}ents. Our 2While closely related to ITQ, we use the the relationship to k-means. term ok-means to emphasize 333000111 686 goal is to find the assignment coefficients B? and the transformation parameters, namely, the rotation R, scaling D, and translation μ, that minimize ?ok-means(B?, R, D,μ) = ?X − μ1T − RDB??f2 (7) = ?X? − RDB??f2 (8) where ?·?f denotes the Frobenius norm, X? ≡ X − μ1T, R iws hceornes ?tr·a?ined to have orthonormal columns≡ ≡(R XTR − μ=1 Im), and D is a positive diagonal matrix. Like k-means, coordinate descent is effective for optimizing (8). We first initialize μ and R, and then iteratively optimize ?ok-means with respect to B?, D, R, and μ: • Optimize B? and D: With straightforward algebraic manipulation, one can show that ?ok-means = ?RTX?−DB??f2 + ?R⊥TX??f2 , (9) where columns of R⊥ span the orthogonal complement of the column-space of R (i.e., the block matrix [R R⊥] is orthogonal). It follows that, given X? and R, we can optimize the first term in (9) to solve for B? and D. Here, DB? is the leastsquares approximation to RTX?, where RTX? and DB? × are m n. Further, the ith row of DB? can only contain earleem men×tsn .fr Fomur t{h−erd,i t,h +e dii} where di = Dii. Wherever tehleem corresponding delement} }o fw RheTrXe? d is positive (negative) we should put a positive (negative) value in DB?. The optimal di is determined by the mean absolute value of the elements in the ith row of RTX?: • • B? ← sgn ?RTX?? (10) D ← Diag?(mroewan?(abs(RTX?))) (11) Optimize R: Using the original objective (8), find R that minimizes ?X? −RA?f2 , subject to RTR = Im, and Am n≡i mDizBes?. ?TXhis i−s equivalent to an Orthogonal Procrustes problem [15], and can be solved exactly using SVD. In particular, by adding p − m rows of zeros to the bottom poaf rDtic, uDlaBr, bb eyc aodmdeins p p× − n. mT rhoewn sR o ifs z square a tnhde orthogoonf aDl ,a DndB can boem eessti pm ×at end. Twhiethn RSV isD s. qBuuatr es ainncde oDrtBho gisdegenerate we are only interested in the first m columns of R; the remaining columns are unique only up to rotation of the null-space.) Optimize μ: Given R, B? and D, the optimal μ is given by the column average of X −RDB?: μ ← mcoeluamn(X−RDB?) (12) 3.2. Approximate nearest neighbor search One application of scalable quantization is ANN search. Before introducing more advanced quantization techniques, we describe some experimental results with ok-means on Euclidean ANN search. The benefits of ok-means for ANN search are two-fold. Storage costs for the centers is reduced to O(log k), from O(k) with k-means. Second, substantial speedups are possible by exploiting fast methods for NN search on binary codes in Hamming space (e.g., [13]). Generally, in terms of a generic quantizer q(·), there are twoG neanteurraalll ways etrom messti omfa ate g ethneer dicis qtaunanceti z beetrw qe(·e)n, ttwheor vectors, v and u [7]. Using the Symmetric quantizer distance (SQD) ?v−u? is approximated by ?q(v)−q(u) ? . Using the Asymmetric quantizer doixsimtanacteed (A byQ ?Dq()v, only one o Uf sthineg tw thoe vectors is quantized, and ?v−u? is estimated as ?v−q(u) ? . vWechtiloer sS iQs qDu might b, aen slightly f?as itse ers t iom compute, vA−QqD(u i)?n-. curs lower quantization errors, and hence is more accurate. For ANN search, in a pre-processing stage, each database entry, u, is encoded into a binary vector corresponding to the cluster center index to which u is assigned. At test time, the queries may or may not be encoded into indices, depending on whether one uses SQD or AQD. In the ok-means model, the quantization of an input x is straightforwardly shown to be qok(x) = μ + RD sgn(RT(x − μ)) . (13) The corresponding m-bit cluster index is sgn(RT(x − μ)). Given two indices, namely b?1 , b?2 ∈ {−1, +1}m,( txhe − symmetric ok-means quantizer distance∈ ∈is { SQDok(b?1, b?2) = ?μ + RDb?1 − μ − RDb?2 ?22 = ?D(b?1 − b?2)?22 .(14) In effect, SQDok is a weighted Hamming distance. It is the sum of the squared diagonal entries of D corresponding to bits where b?1 and b?2 differ. Interestingly, in our experiments with ok-means, Hamming and weighted Hamming distances yield similar results. Thus, in ok-means experiments we simply report results for Hamming distance, to facilitate comparisons with other techniques. When the scaling in ok-means is constrained to be isotropic (i.e., D = αIm for α ∈ R+), then SQDok becomes a constant multiple off othre α αu ∈sua Rl Hamming distance. As discussed in Sec. 5, this isotropic ok-means is closely related to ITQ [5]. The ok-means AQD between a feature vector x and a cluster index b?, is defined as AQDok(x, b?) = ?x −μ − RDb??22 = ?RTx? − Db??22 + ?R⊥Tx??22 , (15) where x? = x−μ. For ANN search, in comparing distances from query x t−o a .d Faotars AetN oNf binary i,n idni ccoesm, ptharei snegc odnisdta tnecrems on the RHS of (15) is irrelevant, since it does not depend on b?. Without this term, AQDok becomes a form of asymmetric Hamming (AH) distance between RTx? and b?. While previous work [6] discussed various ad hoc AH distance measures for binary hashing, interestingly, the optimal AH distance for ok-means is derived directly in our model. 333000111977 1M SIFT, 64−bit encoding (k = 264) lRc@aeR0 . 206148 10 RIoPT1Qk K−Q m( AHeDa)Hn )s (AH)10K Figure 2. Euclidean ANN retrieval results for different methods and distance functions on the 1M SIFT dataset. 3.3. Experiments with ok-means Following [7], we report ANN search results on 1M SIFT, a corpus of 128D SIFT descriptors with disjoint sets of 100K training, 1M base, and 10K test vectors. The training set is used to train models. The base set is the database, and the test points are queries. The number of bits m is typically less than p, but no pre-processing is needed for dimensionality reduction. Rather, to initialize learning, R is a random rotation of the first m principal directions of the training data, and μ is the mean of the data. For each query we find R nearest neighbors, and compute Recall@R, the fraction of queries for which the ground-truth Euclidean NN is found in the R retrieved items. Fig. 2 shows the Recall@R plots for ok-means with Hamming (H) ≈ SQDok and asymmetric Hamming (AH) H≡a mAmQiDngok ( Hd)is t≈an SceQ, vs. ITQ [5] and PQ [7]. The PQ ≡met AhoQdD exploits a more complex asymmetric distance func- tion denoted AD ≡ AQDck (defined in Sec. 6. 1). Note first tthioant od ke-nmoeteadns A improves upon ITQ, with both Hamming and asymmetric Hamming distances. This is due to the scaling parameters (i.e., D) in ok-means. If one is interested in Hamming distance efficient retrieval, ok-means is prefered over ITQ. But better results are obtained with the asymmetric distance function. Fig. 2 also shows that PQ achieves superior recall rates. This stems from its joint encoding of multiple feature dimensions. In ok-means, each bit represents a partition ofthe feature space into two clusters, separated by a hyperplane. The intersection of m orthogonal hyperplanes yields 2m regions. Hence we obtain just two clusters per dimension, and each dimension is encoded independently. In PQ, by comparison, multiple dimensions are encoded jointly, with arbitrary numbers of clusters. PQ thereby captures statistical dependencies among different dimensions. We next extend ok-means to jointly encode multiple dimensions. 4. Cartesian k-means In the Cartesian k-means (ck-means) model, each region center is expressed parametrically as an additive combination of multiple subcenters. Let there be m sets of subcen- Fidg2ure31.Ddep4icton5fCartde?1si2nqdu?4ati5z?3aton 4qDck(da)t= ,?wd i?5134t?h the first (last) two dimensions sub-quantized on the left (right). Cartesian k-means quantizer denoted qck, combines the subquantizations in subspaces, and produces a 4D reconstruction. ters, each with h elements.3 Let C(i) be a matrix whose columns comprise the elements of the ith subcenter set; C(i) ∈ Rp×h. Finally, assume that each cluster center, c, is the∈ sum of exactly one element from each subcenter set: = ?C(i)b(i) m c i?= ?1 , (16) where b(i) ∈ H1/h is a 1-of-h encoding. As a conc∈re Hte example (see Fig. 3), suppose we are given 4D inputs, x ∈ R4, and we split each datum into m = 2 parts: z(1) = ?I2 0? x , and Then, suppose w?e z(2) = ?0 I2? x .(17) qu?antize each part, z(?1) and? z(2) , sepa- × rately. As depicted in Fig. 3 (left and middle), we could use h = 5 subcenters for each one. Placing the corresponding subcenters in the columns of 4 5 matrices C(1) and C(2) , C(1)=?d1d20d2×35d4d5?, C(2)=?d?1d?20d2×?35d?4d?5?, we obtain a model (16) that provides 52 possible centers with which to quantize the data. More generally, the total number of model centers is k = hm. Each center is a member of the Cartesian product of the subcenter sets, hence the name Cartesian k-means. Importantly, while the number of centers is hm, the number of subcenters is only mh. The model provides a super-linear number of centers with a linear number of parameters. The learning objective for Cartesian k-means is ?ck-means(C) =x?∈D{b(mi)}inim=1???x −i?=m1C(i)b(i)??22 where b(i) ∈ H1/h, and C ≡ [C(1), ··· , (18) C(m)] ∈ Rp×mh. [b(1)T, ··· ,b(m)T] If we let bT ≡ then the second sum in (18) can be expressed succinctly as Cb. 3While here we assume a fixed cardinality for all subcenter sets, the model is easily extended to allow sets with different cardinalities. 333000112088 The key problem with this formulation is that the min- imization of (18) with respect to the b(i) ’s is intractable. Nevertheless, motivated by orthogonal k-means (Sec. 3), encoding can be shown to be both efficient and exact if we impose orthogonality constraints on the sets of subcenters. To that end, assume that all subcenters in different sets are pairwise orthogonal; i.e., ∀i,j | i = j C(i)TC(j) = 0h×h .(19) Each subcenter matrix C(i) spans a linear subspace of Rp, and the linear subspaces for different subcenter sets do not intersect. Hence, (19) implies that only the subcenters in C(i) can explain the projection of x onto the C(i) subspace. In the example depicted in Fig. 3, the input features are simply partitioned (17), and the subspaces clearly satisfy the orthogonality constraints. It is also clear that C ≡ [ C(1) C(2)] is block diagonal, Iwtit ihs 2 a ×lso o5 c bleloacrks t,h adte Cnote ≡d D(1) and D(]2 i)s . bTlohec quantization error t×he5re bfolorcek bse,c doemnoeste ?x − Cb?22 = ???zz((12))?−?D0(1) D0(2)? ?b ( 21) ? ?2 = ?????z(1)−D(1)b(1)??2+???z(2)−D(2??)b(2)??2. In words, the squa??zred quantization?? erro??r zis the sum of t??he squared errors on the subspaces. One can therefore solve for the binary coefficients of the subcenter sets independently. In the general case, assuming (19) is satisfied, C can be expressed as a product R D, where R has orthonormal columns, and D is block diagonal; i.e., C = R D where Ra=nd[hRe(n1c),e·C(,i)R=(mR)]i,Dan(di).DW=i⎢t⎡⎣⎢hDs0i.(1≡)Dra0(n2)k. C.(Di)0(.m,i)t⎦⎥ ⎤fo,l(2w0)s that D(i) ∈ Rsi×h and R(i) ∈ Rp×≡sira. Clearly, ? si ≤ p, because of∈ ∈th Re orthogonality ∈con Rstraints. Replacing C(i) with R(i)D(i) in the RHS of (18?), we find ?x−Cb?22 = ?m?z(i)−D(i)b(i)?22+?R⊥Tx?22, (21) i?= ?1 where z(i)≡R(i)Tx, and R⊥is the orthogonal complement of R. This≡ ≡shRows that, with orthogonality constraints (19), the ck-means encoding problem can be split into m independent sub-encoding problems, one for each subcenter set. To find the b(i) that minimizes ??z(i) we perform NN search with z(i) again??st h si-dim vec??tors in D(i) . This entails a cost of O(hsi).? Fortunately, all? the elements of b can be found very efficiently, in O(hs), where s ≡ ? si. If we also include the cost of rotating x to −D(i)b(i)?22, Taocbkml-emt1he.aoAnds um#ceh2nkmrtyeofskm-#lobgeiatknsh,cOo-rm( pOec(2oamkn+spt),hpan)dkO- m(pOecosa(n+mst(hipns)khtesro)m of number of centers, number of bits needed for indices (i.e., log #centers), and the storage cost of representation, which is the same as the encoding cost to convert inputs to indices. The last column shows the costs given an assumption that C has a rank of s ≥ m. obtain each z(i) , the total encoding cost is O(ps + hs), i.e., O(p2+hp). Alternatively, one could perform NN search in p-dim C(i) ’s, to find the b(i) ’s, which costs O(mhp). Table 1 summerizes the models in terms of their number of centers, index size, and cost of storage and encoding. 4.1. Learning ck-means We can re-write the ck-means objective (18) in matrix form with the Frobenius norm; i.e., ?ck-means(R, D, B) = ? X − RDB ?f2 (22) where the columns of X and B comprise the data points and the subcenter assignment coefficients. The input to the learning algorithm is the training data X, the number of subcenter sets m, the cardinality of the subcenter sets h, and an upper bound on the rank of C, i.e., s. In practice, we also let each rotation matrix R(i) have the same number of columns, i.e., si = s/m. The outputs are the matrices {R(i) } and {D(i) } that provide a local minima of (22). Learning begins hwaitth p trohev idneit aia lloizcaatlio mni noimf Ra oanf d(2 D2)., followed by iterative coordinate descent in B, D, and R: • Optimize B and D: With R fixed, the objective is given by (21) where ?R⊥TX?f2 R(i)TX, is constant. Given data pro- jections Z(i) ≡ to optimize for B and D we perform one step oRf k-means for each subcenter set: – Assignment: Perform NN searches for each subcenter set to find the assignment coefficients, B(i) , B(i)← arBg(mi)in?Z(i)− D(i)B(i)?f2 – • Update: D(i)← arDg(mi)in?Z(i)− D(i)B(i)?f2 Optimize R: Placing the D(i) ’s along the diagonal of D, as in (20), and concatenating B(i) ’s as rows of B, [B(1)T, ...,B(m)T], i.e., BT = the optimization of R reduces to the orthogonal Procrustes problem: R ← argRmin?X − RDB?f2. In experiments below, R ∈ Rp×p, and rank(C) ≤ p is unIcnon esxtpraeirniemde. tFso rb high-dimensional adnadta r awnhke(rCe ) ra ≤nk( pX is) ?np, fnostrr efficiency ri th may dbime eusnesfiuoln atol dcoantast wrahiner er anr akn(kC(X). 333000112199 5. Relations and related work As mentioned above, there are close mathematical relationships between ok-means, ck-means, ITQ for binary hashing [5], and PQ for vector quantization [7]. It is instructive to specify these relationships in some detail. Iterative Quantization vs. Orthogonal k-means. ITQ [5] is a variant of locality-sensitive hashing, mapping data to binary codes for fast retrieval. To extract m-bit codes, ITQ first zero-centers the data matrix to obtain X?. PCA is then used for dimensionality reduction, fromp down to m dimensions, after which the subspace representation is randomly rotated. The composition of PCA and the random rotation can be expressed as WTX? where W ∈ Rp×m. ITQ then solves for the m×m rotation mawthriex,r eR W, t ∈hatR minimizes ?ITQ(B?, R) = ?RTWTX?− B??f2 , s.t. RTR = Im×m, (23) where B? ∈ {−1, +1}n×p. ITQ ro∈tate {s− t1h,e+ subspace representation of the data to match the binary codes, and then minimizes quantization error within the subspace. By contrast, ok-means maps the binary codes into the original input space, and then considers both the quantization error within the subspace and the out-of-subspace projection error. A key difference is the inclusion of ?R⊥X? ?f2 in the ok-means objective (9). This is important s ?inRce one can often reduce quantization errors by projecting out significant portions of the feature space. Another key difference between ITQ and ok-means is the inclusion of non-uniform scaling in ok-means. This is important when the data are not isotropic, and contributes to the marked improvement in recall rates in Fig. 2. Orthogonal k-means vs. Cartesian k-means. We next show that ok-means is a special case of ck-means with h = 2, where each subcenter set has two elements. To this end, let C(i) = and let b(i) = be the ith subcenter matrix selection vector. Since b(i) is a 1-of-2 encoding (?10? or ?01?), it follows that: [c(1i) c2(i)], b1(i)c(1i)+ b(2i)c2(i) = [b(1i) b2(i)]T c(1i)+2 c2(i)+ bi?c1(i)−2 c2(i), (24) b1(i) − b2(i) where bi? ≡ ∈ {−1, +1}. With the following setting of t≡he bok-m−e abns parameters, μ=?i=m1c1(i)+2c(2i),andC=?c1(1)−2c2(1),...,c1(m)−2c2(m)?, it should be clear that ?i C(i)b(i) = μ + Cb?, where b? ∈ {−1, +1}m, and b??i is the ith bit of b?, used in (24). Similarly, one can also map ok-means parameters onto corresponding subcenters for ck-means. Thus, there is a 1-to-1 mapping between the parameterizations of cluster centers in ok-means and ck-means for h = 2. The benefits of ok-means are its small number of parameters, and its intuitive formulation. The benefit of the ck-means generalization is its joint encoding of multiple dimensions with an arbitrary number of centers, allowing it to capture intrinsic dependence among data dimensions. Product Quantization vs. Cartesian k-means. PQ first splits the input vectors into m disjoint sub-vectors, and then quantizes each sub-vector separately using a subquantizer. Thus PQ is a special case of ck-means where the rotation R is not optimized; rather, R is fixed in both learning and retrieval. This is important because the sub-quantizers act independently, thereby ignoring intrasubspace statistical dependence. Thus the selection of subspaces is critical for PQ [7, 8]. J e´gou et al. [8] suggest using PCA, followed by random rotation, before applying PQ to high-dimensional vectors. Finding the rotation to minimize quantization error is mentioned in [8], but it was considered too difficult to estimate. Here we show that one can find a rotation to minimize the quantization error. The ck-means learning algorithm optimizes sub-quantizers in the inner loop, but they interact in an outer loop that optimizes the rotation (Sec. 4.1). 6. Experiments 6.1. Euclidean ANN search Euclidean ANN is a useful task for comparing quantization techniques. Given a database of ck-means indices, and a query, we use Asymmetric and Symmetric ck-means quantizer distance, denoted AQDck and SQDck. The AQDck between a query x and a binary index b ≡ ?b(1)T, ...,b(m)T?T, derived in (21), is AQDck(x,b) ?m?z(i)−D(i)b(i)?22+ ?R⊥Tx?22.(25) = i?= ?1 ?z(i)−D(i)b(i)?22is the distance between the ithpro- Here, jection?? of x, i.e., z(i) , ?a?nd a subcenter projection from D(i) selecte?d by b(i) . Give?n a query, these distances for each i, and all h possible values of b(i) can be pre-computed and stored in a query-specific h×m lookup table. Once created, tshtoer lookup qtaubelery i-ss pusecedif itoc compare akullp pda tatbablea.se O points etaot tehde, query. So computing AQDck entails m lookups and additions, somewhat more costly than Hamming distance. The last term on the RHS of (25) is irrelevant for NN search. Since PQ is a special case of ck-means with pre-defined subspaces, the same distances are used for PQ (cf. [7]). The SQDck between binary codes b1 and b2 is given by SQDck(b1,b2) = ?m?D(i)b1(i)− D(i)b2(i)?22.(26) i?= ?1 Since b1(i) and b(2i) are 1-of-?h encodings, an m×h?×h lookup table can be createadre t 1o- sft-ohre e nacllo pairwise msu×bh×-dhis ltaonockeusp. 333000222200 1M SIFT, 64−bit encoding (k = 264) 1M GIST, 64−bit encoding (k = 264) 1B SIFT, 64−bit encoding (k = 264) Re@acl0 .4186021RcIPoTQk0−Q m( SAeDaH)n s (SADH)1K0 .1642081 0RIocP1TkQ −K m (SAeDHa)n s (ASHD1)0K .186420 1 R0IoPc1TkQ −KQm( ASe DaHn) s (SADH1)0K Figure 4. Euclidean NN recall@R (number of items retrieved) based on different quantizers and corresponding distance functions on the 1M SIFT, 1M GIST, and 1B SIFT datasets. The dashed curves use symmetric distance. (AH ≡ AQDok, SD ≡ SQDck, AD ≡ AQDck) While the cost of computing SQDck is the same as AQDck, SQDck could also be used to estimate the distance between the indexed database entries, for diversifying the retrieval results, or to detect near duplicates, for example. Datasets. We use the 1M SIFT dataset, as in Sec. 3.3, along with two others, namely, 1M GIST (960D features) and 1B SIFT, both comprising disjoint sets of training, base and test vectors. 1M GIST has 500K training, 1M base, and 1K query vectors. 1B SIFT has 100M training, 1B base, and 10K query points. In each case, recall rates are averaged over queries in test set for a database populated from the base set. For expedience, we use only the first 1M training points for the 1B SIFT experiments. Parameters. In ANN experiments below, for both ckmeans and PQ, we use m = 8 and h = 256. Hence the number of clusters is k = 2568 = 264, so 64-bits are used as database indices. Using h = 256 is particularly attractive because the resulting lookup tables are small, encoding is fast, and each subcenter index fits into one byte. As h increases we expect retrieval results to improve, but encoding and indexing of a query to become slower. Initialization. To initialize the Di’s for learning, as in kmeans, we simply begin with random samples from the set of Zi’s (see Sec. 4.1). To initialize R we consider the different methods that J e´gou et al. [7] proposed for splitting the feature dimensions into m sub-vectors for PQ: (1) natural: sub-vectors comprise consecutive dimensions, (2) structured: dimensions with the same index modulo 8 are grouped, and (3) random: random permutations are used. For PQ in the experiments below, we use the orderings that produced the best results in [7], namely, the structured ordering for 960D GIST, and the natural ordering for 128D SIFT. For learning ck-means, R is initialized to the identity with SIFT corpora. For 1M GIST, where the PQ ordering is significant, we consider all three orderings to initialize R. Results. Fig. 4 shows Recall@R plots for ck-means and PQ [7] with symmetric and asymmetric distances (SD ≡ PSQQD [7c]k wanitdh A syDm m≡e rAicQ aDncdk) a on tmhee t3r cda dtaissteatns.c Tsh (eS Dhor ≡izontal axainsd represents tQheD number of retrieved items, R, on a log-scale. The results consistently favor ck-means. On 1M GIST, 64−bit encoding (k = 264) lae@Rc0 .261084 1R0Pc Qk −1m(K 321e )a (A nD s )(132) (A D )10K Figure 5. PQ and ck-means results using natural (1), structured (2), and random (3) ordering to define the (initial) subspaces. the high-dimensional GIST data, ck-means with AD significantly outperforms other methods; even ck-means with SD performs on par with PQ with AD. On 1M SIFT, the Recall@ 10 numbers for PQ and ck-means, both using AD, × are 59.9% and 63.7%. On 1B SIFT, Recall@ 100 numbers are 56.5% and 64.9%. As expected, with increasing dataset size, the difference between methods is more significant. In 1B SIFT, each feature vector is 128 bytes, hence a total of 119 GB. Using any method in Fig. 4 (including ckmeans) to index the database into 64 bits, this storage cost reduces to only 7.5 GB. This allows one to work with much larger datasets. In the experiments we use linear scan to find the nearest items according to quantizer distances. For NN search using 10K SIFT queries on 1B SIFT this takes about 8 hours for AD and AH and 4 hours for Hamming distance on a 2 4-core computer. Search can be sped up significantly; using a coarse tienri.tia Sl quantization sapnedd an pin svigenritefidfile structure for AD and AH, as suggested by [7, 1], and using the multi-index hashing method of [13] for Hamming distance. In this paper we did not implement these efficiencies as we focus primarily on the quality of quantization. Fig. 5 compares ck-means to PQ when R in ck-means is initialized using the 3 orderings of [7]. It shows that ckmeans is superior in all cases. Simiarly interesting, it also shows that despite the non-convexity ofthe optimization objective, ck-means learning tends to find similarly good encodings under different initial conditions. Finally, Fig. 6 compares the methods under different code dimensionality. 333000222311 1M GIST, encoding with 64, 96, and 128 bits Rl@ace0 .814206 1 R0cP kQ − m1 69e4216a−8 Knb −itsb 1t69248− bit10K Figure 6. PQ and ck-means results using different number of bits for encoding. In all cases asymmetric distance is used. Table2.Rcokg-nmiteoamPnQCesac(ondksue=r(bkaoc416y=0ko26)n40t2h[eC]IFAR7c598-u1.72096r%a tcesyuingdfferent codebook learning algorithms. 6.2. Learning visual codebooks While the ANN seach tasks above require too many clusters for k-means, it is interesing to compare k-means and ck-means on a task with a moderate number of clusters. To this end we consider codebook learning for bag-ofword√s models [3, 10]. We use ck-means with m = 2 and h = √k, and hence k centers. The main advantage of ck- × here is that finding the closest cluster center is done in O(√k) time, much faster than standard NN search with k-means in O(k). Alternatives for k-means, to improve efficiency, include approximate k-means [14], and hierarchical k-means [12]. Here we only compare to exact k-means. CIFAR-10 [9] comprises 50K training and 10K test images (32 32 pixels). Each image is one of 10 classes (airplane, 3b2i×rd,3 car, cat, )d.e Eera,c dog, frog, sh oonrsee o, ship, laansds etrsu (caki)r.We use publicly available code from Coates et al [2], with changes to the codebook learning and cluster assignment modules. Codebooks are built on 6×6 whitened color image patches. .O Cnoed histogram i sb ucirleta otend 6 per image quadrant, aagnde a linear SVM is applied to 4k-dim feature vectors. Recognition accuracy rates on the test set for different models and k are given in Table 2. Despite having fewer parameters, ck-means performs on par or better than kmeans. This is consistent for different initializations of the algorithms. Although k-means has higher fedility than ckmeans, with fewer parameters, ck-means may be less susceptible to overfitting. Table 2, also compares with the approach of [19], where PQ without learning a rotation is used for clustering features. As expected, learning the rotation has a significant impact on recognition rates, outperforming all three initializations of PQ. mean√s 7. Conclusions We present the Cartesian k-means algorithm, a generalization of k-means with a parameterization of the clus- ter centers such that number of centers is super-linear in the number of parameters. The method is also shown to be a generalization of the ITQ algorithm and Product Quantization. In experiments on large-scale retrieval and codebook learning for recognition the results are impressive, outperforming product quantization by a significant margin. An implementation of the method is available at https : / / github . com/norou z i ckmeans . / References [1] A. Babenko and V. Lempitsky. The inverted multi-index. CVPR, 2012. [2] A. Coates, H. Lee, and A. Ng. An analysis of single-layer networks in unsupervised feature learning. AISTATS, 2011. [3] G. Csurka, C. Dance, L. Fan, J. Willamowski, and C. Bray. Visual categorization with bags of keypoints. ECCV Workshop Statistical Learning in Computer Vision, 2004. [4] T. Ge, K. He, Q. Ke, and J. Sun. Optimized product quantization for approximate nearest neighbor search. CVPR, 2013. [5] Y. Gong and S. Lazebnik. Iterative quantization: A procrustean approach to learning binary codes. CVPR, 2011. [6] A. Gordo and F. Perronnin. Asymmetric distances for binary embeddings. CVPR, 2011. [7] H. J ´egou, M. Douze, and C. Schmid. Product quantization for nearest neighbor search. IEEE Trans. PAMI, 2011. [8] H. J ´egou, M. Douze, C. Schmid, and P. P ´erez. Aggregating local descriptors into a compact image representation. CVPR, 2010. [9] A. Krizhevsky. Learning multiple layers of features from tiny images. MSc Thesis, Univ. Toronto, 2009. [10] S. Lazebnik, C. Schmid, and J. Ponce. Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories. CVPR, 2006. [11] S. P. Lloyd. Least squares quantization in pcm. IEEE Trans. IT, 28(2): 129–137, 1982. [12] D. Nister and H. Stewenius. Scalable recognition with a vocabulary tree. CVPR, 2006. [13] M. Norouzi, A. Punjani, and D. J. Fleet. Fast search in hamming space with multi-index hashing. CVPR, 2012. [14] J. Philbin, O. Chum, M. Isard, J. Sivic, and A. Zisserman. Object retrieval with large vocabularies and fast spatial matching. CVPR, 2007. [15] P. Sch o¨nemann. A generalized solution of the orthogonal procrustes problem. Psychometrika, 3 1, 1966. [16] J. Sivic and A. Zisserman. Video Google: A text retrieval approach to object matching in videos. ICCV, 2003. [17] J. Tenenbaum and W. Freeman. Separating style and content with bilinear models. Neural Comp., 12: 1247–1283, 2000. [18] A. Torralba, K. Murphy, and W. Freeman. Sharing visual features for multiclass and multiview object detection. IEEE T. PAMI, 29(5):854–869, 2007. [19] S. Wei, X. Wu, and D. Xu. Partitioned k-means clustering for fast construction of unbiased visual vocabulary. The Era of Interactive Media, 2012. 333000222422
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